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Course: MATH 180, Spring 2006
School: Mitchell Technical...
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Word Count: 1383

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Class 18.03 1, Feb 8, 2006 Introduction: Geometric view of solving ODE's. Vocabulary: Differential equation; solutions; ordinary; order; general solution, particular solution, initial value; direction field; integral curve; separable equation. Technique: separation of variables. [1] Welcome to 18.03. I hope you've picked up an information sheet and syllabus and a problem set when you came in. Read the...

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MapĂșa Institute of Technology - MATH - 180
18.03 Class 2, February 10, 2006 Numerical Methods [1] The study of differential equations rests on three legs:. Analytic, exact, symbolic methods . Quantitative methods (direction fields, isoclines .) . Numerical methods Even if we can solve symbolicall
MapĂșa Institute of Technology - MATH - 180
18.03 Class 3, Feb 13, 2006 First order linear equations: Models Vocabulary: Coupling constant, system, signal, system response, Models: banks, mixing, cooling, growth and decay. Solution in case the equation is separable; general story deferred to Cla
MapĂșa Institute of Technology - MATH - 180
18.03 Class 4, Feb 15, 2006 First order linear equations: solutions. [1] Definition: A &quot;linear ODE&quot; is one that can be put in the &quot;standard form&quot; _ | | | x' + p(t)x = q(t) | |_| On Monday we looked at the Homogeneous case, x' + p(t) x = 0 . This is se
MapĂșa Institute of Technology - MATH - 180
18.03 Class 5, Feb 17, 2006 Complex Numbers, complex exponential Today, or at least 2006, is the 200th anniversary of the birth of complex numbers. In 1806 papers by Abb\'e Bul\'ee and by Jean-Robert Argand established the planar representation of complex
MapĂșa Institute of Technology - MATH - 180
18.03 Class 6, Feb 21, 2006 Roots of Unity, Euler's formula, Sinusoidal functions [1] Let Roots of unity a&gt;0. Since i^2 = -1 , (+- i sqrt(a)^2 = - a C. C , :Negative real numbers have square roots inAny quadratic polynomial with real coefficients
MapĂșa Institute of Technology - MATH - 180
18.03 Class 7, Feb 22, 2006 Applications of C: Exponential and Sinusoidal input and output: Euler: Re e^cfw_(a+bi)t Im e^cfw_(a+bi)t [1] Integration e^cfw_2t cos(t) ? = = e^cfw_at cos(bt) e^cfw_at sin(bt)Remember how to integrate Use parts twice.
MapĂșa Institute of Technology - MATH - 180
18.03 Class 8, Feb 24, 2006 Autonomous equations I'll use (t,y) today.y' = F(t,y)is the general first order equation y' = g(y) .Autonomous ODE:Eg [Natural growth/decay] Constant growth rate: so y' = k0 y . k0 &gt; 0 means the populuation (if positive) i
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, February 24, 2006Thank you all for your frank responses. Ill try to answer some of the most common confusions. 1. A rst order ODE is autonomous if it has the form y = g (y ). Here y = dy/dt. y is a function of t, so its true t
MapĂșa Institute of Technology - MATH - 180
18.03 Class 9, Feb 27, 2006 Review: Linear v Nonlinear [1] review of linear methods [2] Comment on special features of solutions of linear first order ODEs not shared by nonlinear equations. [1] First Order Linear: x' + p(t) x = q(t)system; input signal;
MapĂșa Institute of Technology - MATH - 180
18.03 Class 11, March 3, 2006 Second order equations: Physical model, characteristic polynomial, real roots, structure of solutions, initial conditions [1] F = ma spring is the basic example. mass Take a spring attached to a wall,dashpot| | | |-&gt; F_ext
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, March 3, 20061. I confused a number of people by dividing through mx + bx + k x = Fext (t) and miraculously getting x + bx + k x = q (t). What I meant to say was that by dividing through you make the coecient of x equal to 1 a
MapĂșa Institute of Technology - MATH - 180
18.03 Class 12, March 6, 2006 Homogeneous constant coefficient linear equations: complex or repeated roots, damping criteria. [1] We are studying equations of the form x&quot; + b x' + k x = 0 (*)which model a mass, dashpot, spring system without external
MapĂșa Institute of Technology - MATH - 180
18.03 Class 13, March 8, 2006 Summary of solutions to homogeneous second order LTI equations; Introduction to inhomogneneous equations. [1] We saw on Monday how to solve x&quot; + bx' + kx = 0.Here is a summary table of unforced system responses. One of three
MapĂșa Institute of Technology - MATH - 180
18.03 Class 14, March 10, 2006 Exponential signals, higher order equations, operators [1] Exponential signals x&quot; + bx' + kx = A e^cfw_rt (*)We want to find some solution. Try for a solution of the form k] b] xp xp' = = B e^cfw_rt xp = B e^cfw_rt :B r e^
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, March 10, 20061. The commonest question concerned the idea and utility of operators. I'll say something now. You can look ahead at the &quot;exponential shift law&quot; if you want, to see one use later. An operator modifies a function
MapĂșa Institute of Technology - MATH - 180
18.03 Class 15, March 13, 2004 Operators: Exponential shift law Undetermined coefficients [1] Operators. D e^cfw_rt so and or D^n e^cfw_rt = = The ERF is based on the following calculation: r e^cfw_rt = rI e^cfw_rtr^n I e^cfw_rt(a_n D^n + . + a_0 I) e^c
MapĂșa Institute of Technology - MATH - 180
18.03 Class 16, March 15, 2006 Frequency response [1] Frequency response: without damping x&quot; + omega_n^2 x = 0 :First recall the Harmonic Oscillator: The spring constant is k = omega_n^2 .Solutions are arbitrary sinusoids with circular frequency the &quot;
MapĂșa Institute of Technology - MATH - 180
18.03 Class 17, March 17, 2006 Application of second order frequency response to AM radio reception with guest appearance by EECS Professor Jeff Lang. [1] The AM radio frequency spectrum is divided into narrow segments which individual stations are requir
MapĂșa Institute of Technology - MATH - 180
18.03 Class 18, March 20, 2006 Review of constant coefficient linear equations: Big example, superposition, and Frequency Response [1] Example. x&quot; + 4x = 0 PLEASE KNOW the solution to the homogeneous harmonic oscillator x&quot; + omega^2 x = 0 are sinusoids of
MapĂșa Institute of Technology - MATH - 180
18.03 Class 20, March 24, 2006 Periodic signals, Fourier series [1] Periodic functions: for example the heartbeat, or the sound of a violin, or innumerable electronic signals. I showed an example of violin and flute. A function f(t) is &quot;periodic&quot; if there
MapĂșa Institute of Technology - MATH - 180
18.03 Class 21, April 3 Fun with Fourier series [1] If f(t) is any decent periodic of period 2pi, it has exactly one expression as (*)f(t) = (a0/2) + a1 cos(t) + a2 cos(2t) + . + b1 sin(t) + b2 sin(2t) + .To be precise, there is a single list of coeffic
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, April 3, 20061. A number of people were confused by my derivation of the Fourier coecients of the function f (t), even, periodic, period 2 , with f (t) = 4 for 0 &lt; t &lt; /2 and f (t) = 0 for /2 &lt; t &lt; . I think the process of exp
MapĂșa Institute of Technology - MATH - 180
18.03 Class 22, April 5 Fourier series and harmonic response [1] My muddy point from the last lecture: I claimed that the Fourier series for f(t) converges wherever \$f\$ is continuous. What does this really say? For example, (pi/4) sq(t) for any value of t
MapĂșa Institute of Technology - MATH - 180
18.03 Class 23, April 7 Step and delta. Two additions to your mathematical modeling toolkit. - Step functions [Heaviside] - Delta functions [Dirac] [1] Model of on/off process: a light turns on; first it is dark, then it is light. The basic model is the H
MapĂșa Institute of Technology - MATH - 180
18.03 Class 24, April 10, 2006 Unit impulse and step responses [1] In real life one often encounters a system with unknown system parameters. If it's a spring/mass/dashpot system you may not know the spring constant, or the mass, or the damping constant.
MapĂșa Institute of Technology - MATH - 180
18.03 Class 25, April 12, 2006 Convolution [1] We learn about a system by studying it responses to various input signals. I claim that the weight function w(t) - the solution to p(D)x = delta(t) with rest initial conditions - contains complete data about
MapĂșa Institute of Technology - MATH - 180
18.03 Lecture 26, April 14 Laplace Transform: basic properties; functions of a complex variable; poles diagrams; s-shift law. [1] The Laplace transform connects two worlds:-| The t domain | | | | t is real and positive | | | | functions f(t) are
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, April 14, 20061. A number of people brought up the point made at the end of Lecture 25, on April 12: how do we know what initial conditions yield the unit step or impulse responses? This is a tricky point and I did not explain
MapĂșa Institute of Technology - MATH - 180
18.03 Class 27, April 17, 2006 Laplace Transform II: inverse transform, t-derivative rule, use in solving ODEs; partial fractions: cover-up method; s-derivative rule. Definition: F(s) = L[f(t)] = integral_cfw_0-^infty f(t) e^cfw_-st dt , Re(s) &gt; 0Rules:
MapĂșa Institute of Technology - MATH - 180
18.03 Class 28, Apr 21 Laplace Transform III: Second order equations; completing the square. Rules: L is linear: L[af(t) + bg(t)] = aF(s) + bG(s)F(s)essentially determinesf(t) = = = = = F(s-a) e^cfw_-as F(s) - F'(s) s F(s) - f(0+) s^2 F(s) - s f
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, April 21, 20061. LTI = Linear, Time Invariant. This is a property of an operator or a system. An operator (which is a rule L that converts one function of time to another one) is linear if L(f + g ) = L(f )+ L(g ) and L(cf ) =
MapĂșa Institute of Technology - MATH - 180
18.03 Class 29, Apr 24 Laplace Transform IV: The pole diagram[1] I introduced the weight function = unit impulse response with the mantra that you know a system by how it responds, so if you let it respond to the simplest possible signal (with the simple
MapĂșa Institute of Technology - MATH - 180
18.03 Class 31, April 28, 2006 First order systems: Introduction [1] There are two fields in which rabbits are breeding like rabbits. Field 1 contains x(t) rabbits, field 2 contains y(t) rabbits. In both fields the rabbits breed at a rate of 3 rabbits per
MapĂșa Institute of Technology - MATH - 180
18.03 Class 32, May 1 Eigenvalues and eigenvectors [1] Prologue on Linear Algebra. [a b ; c d] [x ; y] = x[a ; c] + y[b ; d] :RecallA matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the colu
MapĂșa Institute of Technology - MATH - 180
18.03 Class 33, May 3 Complex or repeated eigenvalues [1] The method for solving u' = Au that we devised on Monday is this:(1) Write down the characteristic polynomial A) p_A(lambda) = det(A - lambda I) = lambda^2 - (tr A)lambda +(det lambda_2 v such tha
MapĂșa Institute of Technology - MATH - 180
18.03 Class 34, May 5 Classification of Linear Phase Portraits The moral of today's lecture: Eigenvalues Rule (usually) A is[1] Recall that the characteristic polynomial of a square matrix p_A(lambda) In the 2x2 case p_A(lambda) where = det(A - lambda
MapĂșa Institute of Technology - MATH - 180
18.03 Class 35, May 8 The companion matrix and its phase portrait; The matrix exponential: initial value problems. [1] We spent a lot of time studying the second order equation x&quot; + bx' + kx = 0and if b and k are nonnegative we interpreted them as the
MapĂșa Institute of Technology - MATH - 180
18.03 Class 36, May 10 Review of matrix exponential Inhomogeneous linear equations [1] Prelude on linear algebra: AB.If A and B are matrices such that the number of columns in A is the same as the number of rows in B , then we can form the &quot;product mat
MapĂșa Institute of Technology - MATH - 180
18.03 Muddy Card responses, May 10, 20061. So what good are exponential matrices? It seems to me that they dont allow us to skip any steps: it looks like you still have to calculate eigenvalues and then eigenvectors, and then use those calculations to co
MapĂșa Institute of Technology - MATH - 180
18.03 Class 37, May 12 Introduction to general nonlinear autonomous systems. [1] Recall that an ODE is &quot;autonomous&quot; if and not on t: x' = g(x) x' depends only on xFor example, I know an island in the St Lawrence River in upstate New York where there are
MapĂșa Institute of Technology - MATH - 180
18.03 Class 38, May 15 Nonlinear systems: Jacobian matrices [1] The Nonlinear Pendulum.The bob of a pendulum is attached to a rod, so it can swing clear around the pivot. This system is determined by three parameters: L m g length of pendulum mass of bo
MapĂșa Institute of Technology - MATH - 180
18.03: Dierential Equations, Spring, 2006 Driving through the dashpotThe Mathlet Amplitude and Phase: Second order considers a spring/mass/dashpot system driven through the spring. If y (t) denotes the displacement of the plunger at the top of the spring
York University - BIOL - 1010
SUBiology 1010 - 2009 Midterm 1 Biochemistry and Cell Biology VERSION A1) You are completing VERSION A. Choose A for this question or you will not be graded. A) Choose THIS ONE. B) C) D) E) 2) A theory is _. A) a poorly supported idea that has little exp
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C r ga ni za er ona Olick t o edit M astti subt it lelst yleC ha nge i n a Compl ex Wor l d3/15/1111Or ganizat ion Change is: Change M anagement is t he conti nuous p r ocess of a l i gni ng an or ganizat ion wit h it s m arket place and doing it mor
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Arkansas - MGMT - 4263
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ChangeRecipients Chapter7 JusticeArticles Toolkit for Organizational Change1RecipientsofChange Includesindividualsatmultipleorganizationallevels! Perceptioniskeytoreactions Reactionscanbepositiveornegative ambivalenceoftencomesfirst Toolkit for
Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Arkansas - MGMT - 4953
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Accounting Measures: Assessing the health of a business is not an easy task since corporations are extremely complex and each one is different from every other one. Therefore, it is not possible to learn a specific process or a limited number of measures
Arkansas - WCOB - 3016
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Arkansas - WCOB - 3016
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